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Giải bài 8.28 trang 53 sách bài tập toán 11 - Kết nối tri thức với cuộc sống

Cho hai biến cố \(A,B\) với \(P\left( A \right) = \frac{1}{4},P\left( {\overline B } \right) = \frac{1}{5},P\left( {A \cup B} \right) = \frac{7}{8}\). Hỏi \(A\) và \(B\) có độc lập hay không?

Tổng hợp đ💯ề thi học kì 2 lớp 11 tất 🏅cả các môn - Kết nối tri thức

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Đề bài

Cho hai biến cố \(A,B\) với \(P\left( A \right) = \frac{1}{4},P\left( {\overline B } \right) = \frac{1}{5},P\left( {A \cup B} \right) = \frac{7}{8}\). Hỏi \(A\) và \(B\) có độc lập hay không? 

Phương pháp giải - Xem chi tiết

Áp dụng quy tắc cộng, nhân xác suất Tính \(P\left( {AB} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right)\) Tính \(P\left( A \right) \cdot P\left( B \right){\rm{\;}}\) So sánh \(P\left( {AB} \right),P\left( A \right).P\left( B \right).\) \(P\left( {AB} \right) = P\left( A \right).P\left( B \right) \Rightarrow \) hai biến cố \(A,B\) độc lập. \(P\left( {AB} \right) \ne P\left( A \right).P\left( B \right) \Rightarrow \) hai biến cố \(A,B\) không độc lập.

Lời giải chi tiết

\(P\left( {AB} \right) = P\left( A \right) + P\left( B \right) - P\left( {A \cup B} \right) = \frac{1}{4} + \left( {1 - \frac{1}{5}} \right) - \frac{7}{8} = \frac{7}{{40}}\).\(P\left( A \right) \cdot P\left( B \right) = \frac{1}{4} \cdot \frac{4}{5} = \frac{1}{5} = \frac{8}{{40}} \ne \frac{7}{{40}} = P\left( {AB} \right){\rm{.\;}}\)Vậy hai biến cố \(A,B\) không độc lập.
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